A proof of space is a proof that a farmer has allocated a portion of their storage in a way that is very difficult to create in real-time but efficient to pre-compute and store on a hard drive. The Chia Proof of Space Construction document goes deeply into the math and implementation considerations to mitigate Hellman's Time - Memory tradeoff problem. A plot is a large set of proofs of space. A harvester can harvest multiple plots on the same machine. A farmer can then control multiple harvesters across many machines to manage the whole "farm."
Farming uses substantially less electricity than Proof of Work for the same unit of security. You can learn more at chiapower.org.